Bar Chart Confirmed

Bar Chart Table Deaths

Data Table of Confirmed and Deaths - Cumulated and Daily Cases

Bar Chart Confirmed per 100,000 Inhabitants

Bar Chart Deaths per 100,000 Inhabitants

World

World

Estimation speed of spread of the Coronavirus with Linear Regression

Exponential Growth and Doubling Time \(T\)

Exponential growth over time can be fitted by linear regression if the logarithms of the case numbers is taken. Generally, exponential growth corresponds to linearly growth over time for the log (to any base) data.

The semi-logorithmic plot with base-10 log scale for the Y axis shows functions following an exponential law \(y(t) = y_0 * a^{t/\tau}\) as straight lines. The time constant \(\tau\) describes the time required for y to increase by one factor of \(a\).

If e.g. the confirmed or death cases are growing in \(t-days\) by a factor of \(10\) the doubling time \(T \widehat{=} \tau\) can be calculated with \(a \widehat{=} 2\) by

\(T[days] = \frac {t[days] * log_{10}(2)} {log_{10}(y(t))-log_{10}(y_0)}\)

with

\(log_{10}(y(t))-log_{10}(y_0) = = log_{10}(y(t))/y_0) = log_{10}(10*y_0/y_0) = 1\)

and doubling time

\(T[days] = t[days] * log_{10}(2) \approx t[days] * 0.30\).

For Spain, Italy, Germany we have had a doubling time up to \(T \approx 9-12 days * 0.3 \approx 2.7 - 4 days !!\).

The doubling time \(T\) and the Forecast is calculated for following selected countries: Austria, France, Germany, Italy, India, South Korea, Spain, United States of America

Germany - Trend with Forecast on a linear scale

Forecast Plot - next 14 days

The plot shows the extreme forecast increase in case of unchecked exponentiell growth. The dark shaded regions show 80% rsp. 95% prediction intervals. These prediction intervals are displaying the uncertainty in forecasts based on the linear regression over the past 7 days.

Comparison Exponential Growth

Germany - Example plot with ~linear slope on a log10 scale

Exponential Growth Evaluation

China and South Korea have slowed down significantly the exponential growth. Therefore, their lines in the chart with the log10 scale no longer have a significant.

Most other countries are still in a phase of more or less unchecked exponentiell growth. For Italy, the reduced exponential growth is reflected in a reduced slope of the cumulated cases.

Virus Spread since mid of January

Check of Forecast Accuracy

The forecast accuracy is checked by using the forecast method for the nine days before the past three days (training data). Subsequent forecasting of the past three days enables comparison with the real data of these days (test data).

The comparison is also an early indicator if the exponential growth is declining. However, possible changes in underreporting (in particular the proportion confirmed / actually infected) requires careful interpretation.

Germany - Forecast Accuracy for past three days

Forecasting with lagged Predictors

The number of confirmed cases can be used as a time delayed predictor of the number of deaths. This will allow comclusions on the time period confirmed to death. More inportant the country specific case fatality rate (CFR, proportion of deaths from confirmed cases) indicates the country specific testing.

Overall a rough conclusion on the country specific proportion of infected to confirmed cases is feasible if the infection fatality rate (IFR, confiremd cases plus all asymptomatic and undiagnosed infections) is assumed to be country independent and the IFR is known (bottom of existing estimates ~ 0.56%, assumption by RKI see https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Modellierung_Deutschland.pdf?__blob=publicationFile ).

Therefore an estimation of the CFR of \(0.06\) \((6\%)\) indicates an underreporting or lack of diagnostic confirmation by a by a factor of ~\(10\). A CFR of \(0.20\) \((20\%)\) indicates an underreporting by a by a factor of ~\(30\). This corresponds to RKI assumption of a underreporting by a factor of \(11-20\) (https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Steckbrief.html).

In the model paper RKI assumes for the

• Incubation period ~ 5-6 days - Day of infection day until symptoms are upcoming)
• Hospitalisation +4 days - Admission to the hospital (if needed) after Incubation Period)
• Average period to death + 11 - if the patient dies, it takes an average of 11 days after admission to the hospital

Depending on the country-specific test frequency (late or early tests), the

*lag_days - time from receipt of the confirmed test result to death, Confirmed to Death, is about 11-13 days.

Note: these methods are also used for example for advertising campaigns. The campaign impact on sales will be some time beyond the end of the campaign, and sales in one month will depend on the advertising expenditure in each of the past few months (see https://otexts.com/fpp3/lagged-predictors.html).

Daily Confirmed and Death Cases

Lag days and Case Fatality Rate (CFR)

Lag days and CFR (proportion of deaths from confirmed cases)

Country	lag_Confirmed	CFR
Germany	13	0.05
Italy	10	0.09
Spain	11	0.06

Forecast residuals indicate quality of fit with Arima model:

ARIMA(Deaths ~ 0 + pdq(d = 0) + lag(Confirmed, lag_days - 2) + lag(Confirmed, lag_days - 1) + lag(Confirmed, lag_days) + lag(Confirmed, lag_days + 1) + lag(Confirmed, lag_days + 2)) .

Daily Deaths depending on lagged Daily Confirmed Cases

Exampla Germany - White Noise of Forecast Residuals